{
  "id": "1511.02101",
  "version": "v1",
  "published": "2015-11-06T14:57:40.000Z",
  "updated": "2015-11-06T14:57:40.000Z",
  "title": "The inclusion of configuration spaces of surfaces in Cartesian products, its induced homomorphism, and the virtual cohomological dimension of the braid groups of S^2 and RP^2",
  "authors": [
    "Daciberg Lima Gonçalves",
    "John Guaschi"
  ],
  "categories": [
    "math.GT",
    "math.AT",
    "math.GR"
  ],
  "abstract": "Let M be a surface, perhaps with boundary, and either compact, or with a finite number of points removed from the interior of the surface. We consider the inclusion i: F\\_n(M) --\\textgreater{} M^n of the nth configuration space F\\_n(M) of M into the n-fold Cartesian product of M, as well as the induced homomorphism i\\_\\#: P\\_n(M) --\\textgreater{} (\\pi\\_1(M))^n, where P\\_n(M) is the n-string pure braid group of M. Both i and i\\_\\# were studied initially by J.Birman who conjectured that Ker(i\\_\\#) is equal to the normal closure of the Artin pure braid group P\\_n in P\\_n(M). The conjecture was later proved by C.Goldberg for compact surfaces without boundary different from the 2-sphere S^2 and the projective plane RP^2. In this paper, we prove the conjecture for S^2 and RP^2. In the case of RP^2, we prove that Ker(i\\_\\#) is equal to the commutator subgroup of P\\_n(RP^2), we show that it may be decomposed in a manner similar to that of P\\_n(S^2) as a direct sum of a torsion-free subgroup L\\_n and the finite cyclic group generated by the full twist braid, and we prove that L\\_n may be written as an iterated semi-direct product of free groups. Finally, we show that the groups B\\_n(S^2) and P\\_n(S^2) (resp. B\\_n(RP^2) and P\\_n(RP^2)) have finite virtual cohomological dimension equal to n-3 (resp. n-2), where B\\_n(M) denotes the full n-string braid group of M. This allows us to determine the virtual cohomological dimension of the mapping class groups of the mapping class groups of S^2 and RP^2 with marked points, which in the case of S^2, reproves a result due to J.Harer.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-11-06T14:57:40.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "configuration space",
      "cartesian product",
      "induced homomorphism",
      "virtual cohomological dimension equal",
      "mapping class groups"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2015arXiv151102101L"
    }
  }
}